A344909 T(n, k) = numerator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Numerators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 2, -3, 0, 2, 0, -13, 0, 4, 21, 0, -17, 0, 2, 0, 177, 0, -7, 0, 4, -77, 0, 2401, 0, -25, 0, 4, 0, -4987, 0, 1123, 0, -29, 0, 8, 1155, 0, -24749, 0, 1499, 0, -11, 0, 2, 0, 718657, 0, -341521, 0, 1157, 0, -74, 0, 4, -4389, 0, 6361429, 0, -495469, 0, 2411, 0, -41, 0, 4

Offset

0,3

Links

Table of n, a(n) for n=0..65.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, 2010. (p. 213-216.)

Alexey Kuznetsov, Expansion of the Riemann Xi Function in Meixner-Pollaczek Polynomials. Canadian Mathematical Bulletin, 51(4) (2008), 561-569.

Dan Romik, Orthogonal polynomial expansions for the Riemann xi function. arXiv:1902.06330 [math.NT], 2019.

Formula

T(n, k) = numerator([x^k] P(n, x), where P(n, x) = i^n*Sum_{k=0..n} (-1)^k* binomial(-3/4 + i*x, k)*binomial(-3/4 - i*x, n - k). The polynomials have the recurrence P(n, x) = (1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))), starting with P(0, x) = 1 and P(1, x) = 2*x.

Example

Triangle starts:
[0]    1;
[1]    0,      2;
[2]   -3,      0,      2;
[3]    0,    -13,      0,       4;
[4]   21,      0,    -17,       0,    2;
[5]    0,    177,      0,      -7,    0,    4;
[6]  -77,      0,   2401,       0,  -25,    0,   4;
[7]    0,  -4987,      0,    1123,    0,  -29,   0,   8;
[8] 1155,      0, -24749,       0, 1499,    0, -11,   0, 2;
[9]    0, 718657,      0, -341521,    0, 1157,   0, -74, 0, 4.

Maple

gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4):
serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n):
row := n -> seq(numer(coeff(coeffz(n), x, k)), k = 0..n):
seq(row(n), n = 0..10);
# Alternative:
CoeffList := p -> numer(PolynomialTools:-CoefficientList(p, x)):
P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else
expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end:
ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);

Mathematica

ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x]
f[n_] := I^n Sum[(-1)^k Binomial[-3/4 + I*x, k] Binomial[-3/4 - I*x, n-k], {k, 0, n}] // ForceSimpl;
row[n_] := CoefficientList[f[n], x] // Numerator;
Table[row[n], {n, 0, 10}] // Flatten

Crossrefs

Cf. A344910 (denominators).

Cf. A067002 (column 0), A001316 (column n).

Sequence in context: A065861 A329393 A336207 * A126832 A068908 A226192

Adjacent sequences: A344906 A344907 A344908 * A344910 A344911 A344912

Keyword

sign,tabl,frac

Author

Peter Luschny, Jul 08 2021

Status

approved